# IB DP Maths HL: Past Years Question Bank with Solution – Paper 2

### Paper 2

Instructions to candidates

• Write your session number in the boxes above.
• Do not open this examination paper until instructed to do so.
• A graphic display calculator is required for this paper.
• Section B: answer all questions in the answer booklet provided. Fill in your session number on the front of the answer booklet, and attach it to this examination paper and your cover sheet using the tag provided.
• Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures.
• A clean copy of the mathematics HL and further mathematics HL formula booklet is required for this paper.
• The maximum mark for this examination paper is [100 marks].
• Time: 120 minutes

Topic 7 – Option: Statistics and probability

• Topic 7.1
• Cumulative distribution functions for both discrete and continuous distributions.
• Geometric distribution.
• Negative binomial distribution.
• Probability generating functions for discrete random variables.
• Using probability generating functions to find mean, variance and the distribution of the sum of $$n$$ independent random variables.
• Topic 7.2
• Linear transformation of a single random variable.
• Mean of linear combinations of $$n$$ random variables.
• Variance of linear combinations of $$n$$ independent random variables.
• Expectation of the product of independent random variables.
• Topic 7.3
• Unbiased estimators and estimates.
• Comparison of unbiased estimators based on variances.
• $${\bar X}$$ as an unbiased estimator for $$\mu$$ .
• $${S^2}$$ as an unbiased estimator for $${\sigma ^2}$$ .
• Topic 7.4
• A linear combination of independent normal random variables is normally distributed. In particular, $$X{\text{ ~ }}N\left( {\mu ,{\sigma ^2}} \right) \Rightarrow \bar X{\text{ ~ }}N\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)$$ .
• The central limit theorem.
• Topic 7.5
• Confidence intervals for the mean of a normal population.
• Topic 7.6
• Null and alternative hypotheses, $${H_0}$$ and $${H_1}$$ .
• Significance level.
• Critical regions, critical values, $$p$$-values, one-tailed and two-tailed tests.
• Type I and II errors, including calculations of their probabilities.
• Testing hypotheses for the mean of a normal population.
• Topic 7.7
• Introduction to bivariate distributions.
• Covariance and (population) product moment correlation coefficient $$\rho$$.
• Proof that $$\rho = 0$$ in the case of independence and $$\pm 1$$ in the case of a linear relationship between $$X$$ and $$Y$$.
• Definition of the (sample) product moment correlation coefficient $$R$$ in terms of n paired observations on $$X$$ and $$Y$$.Its application to the estimation of $$\rho$$.
• Informal interpretation of $$r$$, the observed value of $$R$$. Scatter diagrams.
• Topics based on the assumption of bivariate normality: use of the $$t$$-statistic to test the null hypothesis $$\rho = 0$$ .
• Topics based on the assumption of bivariate normality: knowledge of the facts that the regression of $$X$$ on $$Y$$ ($${E\left. {\left( X \right)} \right|Y = y}$$) and $$Y$$ on $$X$$ ($${E\left. {\left( Y \right)} \right|X = x}$$) are linear.
• Topics based on the assumption of bivariate normality: least-squares estimates of these regression lines (proof not required).
• Topics based on the assumption of bivariate normality: the use of these regression lines to predict the value of one of the variables given the value of the other.

Topic 8 – Option: Sets, relations and groups

• Topic 8.1
• Finite and infinite sets. Subsets.
• Operations on sets: union; intersection; complement; set difference; symmetric difference.
• De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).
• Topic 8.2
• Ordered pairs: the Cartesian product of two sets.
• Relations: equivalence relations; equivalence classes.
• Topic 8.3
• Functions: injections; surjections; bijections.
• Composition of functions and inverse functions.
• Topic 8.4
• Binary operations.
• Operation tables (Cayley tables).
• Topic 8.5
• Binary operations: associative, distributive and commutative properties.
• Topic 8.6
• The identity element $$e$$.
• The inverse $${a^{ – 1}}$$ of an element $$a$$.Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.
• Proofs of the uniqueness of the identity and inverse elements.
• Topic 8.7
• The definition of a group $$\left\{ {G, * } \right\}$$ .
• The operation table of a group is a Latin square, but the converse is false.
• Abelian groups.
• Topic 8.8
• Example of groups: $$\mathbb{R}$$, $$\mathbb{Q}$$, $$\mathbb{Z}$$ and $$\mathbb{C}$$ under addition.
• Example of groups: integers under addition modulo $$n$$.
• Example of groups: non-zero integers under multiplication, modulo $$p$$, where $$p$$ is prime.
• Symmetries of plane figures, including equilateral triangles and rectangles.
• Invertible functions under composition of functions.
• Topic 8.9
• The order of a group.
• The order of a group element.
• Cyclic groups.
• Generators.
• Proof that all cyclic groups are Abelian.
• Topic 8.10
• Permutations under composition of permutations.
• Cycle notation for permutations.
• Result that every permutation can be written as a composition of disjoint cycles.
• The order of a combination of cycles.
• Topic 8.11
• Subgroups, proper subgroups.
• Use and proof of subgroup tests.
• Definition and examples of left and right cosets of a subgroup of a group.
• Lagrange’s theorem.
• Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)
• Topic 8.12
• Definition of a group homomorphism.
• Definition of the kernel of a homomorphism.
• Proof that the kernel and range of a homomorphism are subgroups.
• Proof of homomorphism properties for identities and inverses.
• Isomorphism of groups.
• The order of an element is unchanged by an isomorphism.